I sometimes have them

You can use DVDShrink to make a copy of one of your exisiting discs, but tell it to region code that copy for region two prior to encoding. That's the easiest way I can think of doing it.

Regional settings are set at the disc authoring/mastering level and not on the video itself, so simply having a clip of something and burning it isn't going to help. You'll need to get ahold of an honest to goodness region 2 DVD to test out your player.